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(Slate)   1+2+3+4+5+...=-1/12. No so fast "Unlikely"--this is used several areas of physics. Farker and math dealer, The Bad Astronomer, explains. Subby always thought something was fundamentally wrong with the universe   (slate.com) divider line 117
    More: Strange, Phil Plait, physics, universe, infinite series, Death from the Skies, Bad Astronomy, maths  
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6211 clicks; posted to Geek » on 19 Jan 2014 at 8:40 PM (36 weeks ago)   |  Favorite    |   share:  Share on Twitter share via Email Share on Facebook   more»



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2014-01-19 05:02:46 PM
I asked a professor of mathematics about this and here's his reaction:

This just goes to show the dangers of the "axiom of choice". By clever choices of where you add/subtract the numbers you can show anything with an infinite series. ( Which shows that choice is an illusion and by extension so is free will. Welcome to your meaningless life.:-) ) In this case he chose to not finish adding one series before sifting the terms of the next series in there.

As a rule we do not rearrange the terms of infinite series because unless they are absolutely convergent Riemann showed that you can sum to any real number.

String theory is mental masturbation for mathematicians and physicists.
 
2014-01-19 05:09:11 PM
Explain how any sum of integers can be a non-integer?

If you want to put 1 over those integers, then sure.
 
2014-01-19 05:13:43 PM

b0rscht: Explain how any sum of integers can be a non-integer?

If you want to put 1 over those integers, then sure.


It can if you throw out all of the normal rules of math and substitute them with new rules.  Those new rules may be internally consistent with themselves, but that doesn't mean that they'll play nice with the normal accepted rules.

It's not even that the come out with a non-integer sum, but that they come out with a negative non-integer sum that's really brain-breaking.
 
2014-01-19 05:14:21 PM

vygramul: String theory is mental masturbation for mathematicians and physicists.


Yeah, but without those people would be wandering the streets with cardboard signs reading "Will derive 12 dimensions of q branes for food."
 
2014-01-19 05:27:37 PM
For an alternative view, here's a high-energy physicist on the subject:

In an annoyingly all-day meeting so I don't have time to get into it in too much detail. It comes down to a technique called Zeta function regularization that dates back to Euler. Some details on wikipedia:

https://en.wikipedia.org/wiki/Zeta_function_regularization

The concept of regularization (renormalization in physics-speak) is a common technique that is effectively required of all quantum field theories (something that string theory must be to be a physics theory, someday). The gist is that the math introduces infinities (divergences) in places that are not at all physical; you introduce a cut-off or some other parameter to avoid this, assuming that you can also demonstrate that everything is mathematically consistent.
 
2014-01-19 06:26:50 PM
Mental masturbation is right.  My reaction to that explanation is this:  We're not talking about 3 equations.  We're not even talking about 2.  We're talking about 1 equation, which includes no negatives.  Your argument is interesting, but bullshiat in relation to the question at hand.
 
2014-01-19 06:39:03 PM

Benevolent Misanthrope: Mental masturbation is right.  My reaction to that explanation is this:  We're not talking about 3 equations.  We're not even talking about 2.  We're talking about 1 equation, which includes no negatives.  Your argument is interesting, but bullshiat in relation to the question at hand.


It's really similar to the algebra used to show that 0.99999... = 1.

X=0.9999999...
10X=9.999999...
10X-X=9.999999...-0.999999...
9X=9
X=1
 
2014-01-19 07:34:27 PM
Pick your postulates, you can do anything.  I thought it would have been more entertaining if they'd done it in something besides base 10.
 
2014-01-19 08:05:04 PM

vygramul: Benevolent Misanthrope: Mental masturbation is right.  My reaction to that explanation is this:  We're not talking about 3 equations.  We're not even talking about 2.  We're talking about 1 equation, which includes no negatives.  Your argument is interesting, but bullshiat in relation to the question at hand.

It's really similar to the algebra used to show that 0.99999... = 1.

X=0.9999999...
10X=9.999999...
10X-X=9.999999...-0.999999...
9X=9
X=1


How is it similar to that? This bullshiat result is contradictory, unlike that one. If you are telling me that adding purely positive numbers can result in a negative number, you've contradicted a fundamental basis of math. It's not possible.
 
2014-01-19 08:32:26 PM

DamnYankees: vygramul: Benevolent Misanthrope: Mental masturbation is right.  My reaction to that explanation is this:  We're not talking about 3 equations.  We're not even talking about 2.  We're talking about 1 equation, which includes no negatives.  Your argument is interesting, but bullshiat in relation to the question at hand.

It's really similar to the algebra used to show that 0.99999... = 1.

X=0.9999999...
10X=9.999999...
10X-X=9.999999...-0.999999...
9X=9
X=1

How is it similar to that? This bullshiat result is contradictory, unlike that one. If you are telling me that adding purely positive numbers can result in a negative number, you've contradicted a fundamental basis of math. It's not possible.


I'm not saying that at all. I'm saying when you deal with infinites, you get interesting results, and you have to be careful what you are doing.
 
2014-01-19 08:40:23 PM

vygramul: I'm not saying that at all. I'm saying when you deal with infinites, you get interesting results, and you have to be careful what you are doing.


While i agree with you in principle, the proof that .999...=1 doesn't really required infinites at all and can be proved other ways than the way you did it (by subtracting on infinite from another).
 
2014-01-19 08:44:12 PM
42

/so long and thanks....
 
2014-01-19 08:51:52 PM
This constipated mathematician clearly needs to work it out with a pencil.
 
2014-01-19 08:52:42 PM
1 plus 1 equals three*


*assuming sufficiently high values of "one".
 
2014-01-19 08:54:03 PM
It uses a trick that's against the rules, so strictly speaking it doesn't work.... I had my suspicions about the method used in the video, but suppressed them.

As "silly" as I want to point out this all was, at least the guy is honest about it.  So my panties remain unbunched.

DamnYankees: vygramul: I'm not saying that at all. I'm saying when you deal with infinites, you get interesting results, and you have to be careful what you are doing.

While i agree with you in principle, the proof that .999...=1 doesn't really required infinites at all and can be proved other ways than the way you did it (by subtracting on infinite from another).


1/3 = 0.333..
2/3 = 0.666...
3/3 = ?
 
2014-01-19 09:06:09 PM

vygramul: String theory is mental masturbation for mathematicians and physicists.


Maybe, maybe not.  But ultimately irrelevant.  This is also used in areas of physics that well understood and well verified by this thing called experimentation such as the Casimir effect.

And I sure hope no one here is going to claim to be a better math whiz than Euler.

But think of it this way, it is utterly impossible to actually add every single natural number together and the series of partial sums does not converge on anything.    The result of an impossible summation that has no real meaning in the real world can't help but be bizarre.
 
2014-01-19 09:07:12 PM

TheMysteriousStranger: it is utterly impossible to actually add every single natural number together and the series of partial sums does not converge on anything.


Why not.
 
2014-01-19 09:09:19 PM
Math dealers blow your mind far worse than any meth dealer and they don't toss them in jail.
 
2014-01-19 09:10:33 PM
There are times when I think Plait is a great popularizer of science, then there are times when I read him and facepalm.  The Andromeda galaxy thing was one of the first, this is the second category. Summary of his post "isn't renormalization weird?  It's SOOOO weeeeird!!! But it works, so all yall who said I was wrong are wronger!"  I guess I should be thankful he didn't write a whole book on it, like some people have with how "weird" Heisenberg's relation is.

Also, when he links to string crap as a 'useful' example, he ain't helpin' his case.
 
2014-01-19 09:11:22 PM
FTFA: Or, they can be self-consistent, totally logical, and not useful.

As soon as I read this line, I came here to make fun of the Axiom of Choice, but I see it was taken care of in the very Boobies.

Fark, I am impressed.
 
2014-01-19 09:11:45 PM

Any Pie Left: 1 plus 1 equals three*


*assuming sufficiently high values of "one".


I was told that one and one and one is three.
 
2014-01-19 09:18:53 PM

macross87: 42

/so long and thanks....


six by nine... by nine?
 
2014-01-19 09:19:26 PM
Still playing games with Grandi's series, huh.

upload.wikimedia.org

or 1-1+1-1+1-1.....

Well, this is the way >I< group them for addition:

Take, all the positive +1's, and sum them, take all the negative 1's and sum them, sum together

so you have (upload.wikimedia.org *1) + (upload.wikimedia.org * -1)
aka  ∞ -∞
aka whatever the fark you want.
 
2014-01-19 09:23:12 PM
Here's a nice, direct link to a nice, sensible refutation. As a bonus (to me), it's from a former classmate. (Hi, Erann/Ron!)
 
2014-01-19 09:24:24 PM

DamnYankees: TheMysteriousStranger: it is utterly impossible to actually add every single natural number together and the series of partial sums does not converge on anything.

Why not.


You can keep adding terms and you never get the job done.  If you added 1+2+3+...+googolplex factorial and get a result you have not computed the sum in question.   If you live an eternity and you never stop adding you will never get a result.   And unlike some sums like one half + one quarter + one eighth + ..., the partial sums don't clearly converge on an "answer" which that case would be one.  So in the normal sense which we usually work it makes not sense to say that 1+2+3+4+5+... has an answer.   But something called the zeta function which is usually in many areas zeta of -1 has a value of -1/12 and it is equivalent to 1+2+3+....

Wikipedia actually has an articles with references:

1 + 2 + 3 + 4 + ⋯

Riemann zeta function
 
2014-01-19 09:25:35 PM

DamnYankees: vygramul: I'm not saying that at all. I'm saying when you deal with infinites, you get interesting results, and you have to be careful what you are doing.

While i agree with you in principle, the proof that .999...=1 doesn't really required infinites at all and can be proved other ways than the way you did it (by subtracting on infinite from another).


I didn't say it was the only way.
 
2014-01-19 09:27:24 PM

TheMysteriousStranger: But something called the zeta function which is usually in many areas zeta of -1 has a value of -1/12 and it is equivalent to 1+2+3+....


But something called the zeta function which is useful in many areas, zeta of -1 has a value of -1/12...

/must not hit "add comment" so fast
//will never learn
 
2014-01-19 09:28:32 PM

vygramul: DamnYankees: vygramul: I'm not saying that at all. I'm saying when you deal with infinites, you get interesting results, and you have to be careful what you are doing.

While i agree with you in principle, the proof that .999...=1 doesn't really required infinites at all and can be proved other ways than the way you did it (by subtracting on infinite from another).

I didn't say it was the only way.


I know, I was expanding on your point, not refuting it.
 
2014-01-19 09:29:33 PM
I was told there would not be math.
 
2014-01-19 09:32:25 PM

DamnYankees: vygramul: DamnYankees: vygramul: I'm not saying that at all. I'm saying when you deal with infinites, you get interesting results, and you have to be careful what you are doing.

While i agree with you in principle, the proof that .999...=1 doesn't really required infinites at all and can be proved other ways than the way you did it (by subtracting on infinite from another).

I didn't say it was the only way.

I know, I was expanding on your point, not refuting it.


Roger that.
 
2014-01-19 09:36:34 PM
"It uses a trick that's against the rules, so strictly speaking it doesn't work."

farm4.staticflickr.com
 
2014-01-19 09:38:52 PM

DamnYankees: How is it similar to that? This bullshiat result is contradictory, unlike that one. If you are telling me that adding purely positive numbers can result in a negative number, you've contradicted a fundamental basis of math. It's not possible.


I realize that some people are not able to think abstractly enough to be able to accept that something that they were taught in third grade might not be the absolute truth they always believed.  Some people's brains just aren't pliable enough.  But I'll try to explain.

First of all, no one is saying that, under the normal rules of arithmetic, that 1+2+3+4+5+... = -1/12.  1+2+3+4+5 is infinite and there is no other answer.

But not all mathematicians play by those rules.  Sometimes mathematicians, as a hypothetical exercise, will change the rules.  They, for example, can define an operation that is a lot like addition, but is not exactly like it.  Maybe this altered form of addition is the same as regular addition when used in typical cases (1+1=2), but for werid cases like infinite sums it's defined to act act differently than addition.

This altered form of addition is not really addition according to the normal rules, and no one is saying that it is.  But, if we all agree, as a hypothetical, to use a different set of rules, the operation that is kind of like addition can, in fact, result in a negative number.  (Or, rather, an entity that is something like a negative number.)

Furthermore, arithmetic done with these hypothetical sets of rules can be useful.

Don't believe me?  You don't have to look at group theory, transfinite theory or any other loopy branch of mathematics.  Look at your desktop computer.  Try running this little C program:

#include <stdio.h>
int main(void) {
    int c;
    c = 2000000000;
    c += 2000000000;
    printf("%d\n", c);
    return 0;
}


It adds two billion to two billion and tries to store it in a 32-bit two's-compliement integer.  I get -294967296.  Adding two positives yields a negative.

How is that possible?  Because 32-bit two's complement "integers" aren't really integers, and the C addition operator isn't really addition.  But you can't deny that "integer" "arithmetic" in C more or less the same as integer artihmetic, as long as the integers aren't too big.  Nor can you deny that it's useful, even though it's not actually integers and not really arithmetic.
 
2014-01-19 09:44:24 PM

aerojockey: First of all, no one is saying that, under the normal rules of arithmetic, that 1+2+3+4+5+... = -1/12.  1+2+3+4+5 is infinite and there is no other answer.


Ok so we're done.

aerojockey: But not all mathematicians play by those rules.  Sometimes mathematicians, as a hypothetical exercise, will change the rules.


So it's not math. I mean, how is this different than just defining your own words? It's meaningful blather.
 
2014-01-19 09:49:06 PM
This is the kind of shiat that makes people not trust scientists.
 
2014-01-19 09:55:56 PM

DamnYankees: aerojockey: But not all mathematicians play by those rules. Sometimes mathematicians, as a hypothetical exercise, will change the rules.

So it's not math. I mean, how is this different than just defining your own words? It's meaningful blather.


Did you reach the end of my post?  By the exact same logic, computers (mostly) don't do math.  Adding two integers is not addition, because you can add two positives on a computer and get a negative.

So do you scream a programmers who say they're adding two numbers?  Do you say, "you're not really adding you dolt"?  Do you demand that programmers and computer scientists create their own words for "addition"?
 Why do you accept that computer scientists can use words like addition for things that aren't addition, but mathematicians can't?


I can answer that: your brain isn't that pliable.
 
2014-01-19 09:59:55 PM

DamnYankees: So it's not math. I mean, how is this different than just defining your own words? It's meaningful blather.


That would be the case if that blather didn't have actual real-world applications. So obviously it's not blather.
 
2014-01-19 09:59:58 PM

DamnYankees: So it's not math. I mean, how is this different than just defining your own words? It's meaningful blather.


"Math" is not some singular, absolute thing.  We make rules, and then we play by them.  And there are many different common rulesets.  Take the Axiom of Choice.  Or take its negation.  Or take some other sort of axiom.  Maybe a few large cardinal axioms?  Or go back to naive set theory.  etc...

So yes, it's still math.
 
2014-01-19 10:02:37 PM
Here's the original video that he's likely talking about: http://www.youtube.com/watch?v=w-I6XTVZXww along with 2 proofs.
 
2014-01-19 10:23:33 PM

RaisingKane: DamnYankees: So it's not math. I mean, how is this different than just defining your own words? It's meaningful blather.

"Math" is not some singular, absolute thing.  We make rules, and then we play by them.  And there are many different common rulesets.  Take the Axiom of Choice.  Or take its negation.  Or take some other sort of axiom.  Maybe a few large cardinal axioms?  Or go back to naive set theory.  etc...

So yes, it's still math.


Oh for fark's sake, what's next then, imaginary numbers?  And then what, you'll find some sort of use for these imagined up numbers?  Get outta here with that stuff.
 
2014-01-19 10:26:58 PM

vygramul: Benevolent Misanthrope: Mental masturbation is right.  My reaction to that explanation is this:  We're not talking about 3 equations.  We're not even talking about 2.  We're talking about 1 equation, which includes no negatives.  Your argument is interesting, but bullshiat in relation to the question at hand.

It's really similar to the algebra used to show that 0.99999... = 1.

X=0.9999999...
10X=9.999999...
10X-X=9.999999...-0.999999...
9X=9
X=1


Sigh.  The connection was reset and Fark ate my post.  Let's sum up.

upload.wikimedia.org

That's the sum of an infinite geometric series, like a repeating decimal.  a/(1-r) where a is the thing that repeats, and r is the common ratio.  So

a = .9
r = .1

and .999999... = .9/(1-.1)

The rest of the proof is left as an exercise for the reader.
 
2014-01-19 10:33:38 PM

theorellior: DamnYankees: So it's not math. I mean, how is this different than just defining your own words? It's meaningful blather.

That would be the case if that blather didn't have actual real-world applications. So obviously it's not blather.


Actually, it would still be the case even if it didn't have real-world applications. Quite a lot of mathematics has limited or no real-world use. (Has anyone discovered a real-world use for Fermat's Last Theorem yet?)

What's funny is that sometimes a branch of math seems to be a useless theoretical exercise at first but turns out to have real-world applications later. Off hand I can think of quaternion arithmetic and group theory. (Both of which allow for redefinitions of standard arithmetic, BTW.)

Some people have unilaterally decided for themselves that anything that doesn't adhere to the rules of arithmetic they learned in grade school isn't "mathematics", as if they invented the term. But no matter how much they protest, any abstract self-consistent set of rules falls under mathematics, whether it's useful or not, and if it co-opts terminology from other branches.
 
2014-01-19 10:33:43 PM
m0vie.files.wordpress.com

1+2+1+1
 
2014-01-19 10:34:14 PM

RaisingKane: DamnYankees: So it's not math. I mean, how is this different than just defining your own words? It's meaningful blather.

"Math" is not some singular, absolute thing.  We make rules, and then we play by them.  And there are many different common rulesets.  Take the Axiom of Choice.  Or take its negation.  Or take some other sort of axiom.  Maybe a few large cardinal axioms?  Or go back to naive set theory.  etc...

So yes, it's still math.


You are mixing up the meaning of 'definitions' and 'axioms'. You can play around with axioms all you want but changing definitions is the one thing you don't do unless you've got a damned good reason for it. If everyone works with different definitions you have no ground for comparing what you've done against other stuff. To take your own example, suppose that one person proves the Axiom of choice is independent of ZF, another person with another definition could go and prove the opposite, and nobody could make any sense of what was what.

This youtube stuff is all nonsense. The sum of the natural numbers is plus infinity.
 
2014-01-19 11:04:14 PM

vygramul: DamnYankees: vygramul: Benevolent Misanthrope: Mental masturbation is right.  My reaction to that explanation is this:  We're not talking about 3 equations.  We're not even talking about 2.  We're talking about 1 equation, which includes no negatives.  Your argument is interesting, but bullshiat in relation to the question at hand.

It's really similar to the algebra used to show that 0.99999... = 1.

X=0.9999999...
10X=9.999999...
10X-X=9.999999...-0.999999...
9X=9
X=1

How is it similar to that? This bullshiat result is contradictory, unlike that one. If you are telling me that adding purely positive numbers can result in a negative number, you've contradicted a fundamental basis of math. It's not possible.

I'm not saying that at all. I'm saying when you deal with infinites, you get interesting results, and you have to be careful what you are doing.


Can we not start this shiat again? I'm no mathematician but I know the difference between numbers and numerals and know that 0.333.. or 0.999... has absolutely nothing to do with "infinites"

Just because it takes an "infinite" number of *numerals* to write something in base 10 doesn't mean the number itself has anything to do with "infinites"
 
2014-01-19 11:18:08 PM

jonny_q: vygramul: DamnYankees: vygramul: Benevolent Misanthrope: Mental masturbation is right.  My reaction to that explanation is this:  We're not talking about 3 equations.  We're not even talking about 2.  We're talking about 1 equation, which includes no negatives.  Your argument is interesting, but bullshiat in relation to the question at hand.

It's really similar to the algebra used to show that 0.99999... = 1.

X=0.9999999...
10X=9.999999...
10X-X=9.999999...-0.999999...
9X=9
X=1

How is it similar to that? This bullshiat result is contradictory, unlike that one. If you are telling me that adding purely positive numbers can result in a negative number, you've contradicted a fundamental basis of math. It's not possible.

I'm not saying that at all. I'm saying when you deal with infinites, you get interesting results, and you have to be careful what you are doing.

Can we not start this shiat again? I'm no mathematician but I know the difference between numbers and numerals and know that 0.333.. or 0.999... has absolutely nothing to do with "infinites"

Just because it takes an "infinite" number of *numerals* to write something in base 10 doesn't mean the number itself has anything to do with "infinites"


I never thought a peripherally-related comment would cause this much consternation.
 
2014-01-19 11:37:52 PM
That headline gave me a headache.
 
2014-01-19 11:42:16 PM
This of course proves that infinity can't exist in real life. It was just invented as a short cut to handle monstrously large numbers and sort of grew into infinity being treated like a real thing.

Numbers aren't real life.  They are reifications.
 
2014-01-19 11:48:29 PM

vygramul: Benevolent Misanthrope: Mental masturbation is right.  My reaction to that explanation is this:  We're not talking about 3 equations.  We're not even talking about 2.  We're talking about 1 equation, which includes no negatives.  Your argument is interesting, but bullshiat in relation to the question at hand.

It's really similar to the algebra used to show that 0.99999... = 1.

X=0.9999999...
10X=9.999999...
10X-X=9.999999...-0.999999...
9X=9
X=1


1/3 = 0.3333...

1/3 + 1/3 + 1/3 = 0.3333...  + 0.3333...  + 0.3333...  = 0.9999... = 3/3 = 1
 
2014-01-20 12:03:22 AM
A system of infinite size can't be reduced to anything. It's infinity all the way down.
 
2014-01-20 12:25:39 AM
Cauchy's smaller friend

planettracy.com

Incidentally

i.chzbgr.com
 
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